Group #11786177 — subgroups

Subgroups — #11786177

Order 1152.

450 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 290 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄³×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₉⋊C₈)	C₂	1	6	yes	maximal
3	#69394	—	3	1	no	maximal
4	C₂×C₄²×C₉	C₄	1	3	yes	—
5	C₂×C₄²×C₉	C₄	1	1	yes	characteristic
6	C₂×C₄²×C₉	C₂²	1	3	yes	—
7	C₂²×C₉⋊C₈	C₂²	1	4	yes	—
8	C₄⋊(C₉⋊C₈)	—	2	12	no	—
9	C₃×C₄³	D₃	1	1	yes	characteristic
10	C₂×C₄⋊(C₃⋊C₈)	—	3	6	no	—
11	C₂²×C₄×C₉	C₂×C₄	1	3	yes	—
12	C₂²×C₄×C₉	C₂×C₄	1	3	yes	—
13	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
14	C₄²×C₉	—	2	6	no	—
15	C₄²×C₉	C₈	1	4	yes	complemented
16	C₄²×C₉	D₄	1	6	yes	—
17	C₄²×C₉	Q₈	1	6	yes	—
18	C₂×C₉⋊C₈	—	2	12	no	—
19	#1842	—	9	1	no	Sylow
20	C₂×C₃×C₄²	Dic₃	1	3	yes	—
21	C₂×C₃×C₄²	Dic₃	1	1	yes	characteristic
22	C₂×C₃×C₄²	C₂×D₃	1	3	yes	—
23	C₂²×C₃⋊C₈	—	3	4	no	—
24	C₄⋊(C₃⋊C₈)	—	6	12	no	—
25	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₉	—	2	6	no	—
27	C₂×C₄×C₉	C₄⋊C₄	1	6	yes	—
28	C₂×C₄×C₉	C₄⋊C₄	1	6	yes	—
29	C₂×C₄×C₉	C₂×C₈	1	6	yes	—
30	C₂×C₄×C₉	M₁₆	1	6	yes	—
31	C₂×C₄×C₉	C₂×D₄	1	3	yes	—
32	C₂×C₄×C₉	C₂×Q₈	1	3	yes	—
33	C₉⋊C₈	—	4	4	no	—
34	C₂×C₄⋊C₈	—	9	6	no	—
35	C₄³	D₉	1	1	yes	characteristic
36	C₂²×C₃×C₄	C₂×Dic₃	1	3	yes	—
37	C₂²×C₃×C₄	C₂×Dic₃	1	3	yes	—
38	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
39	C₃×C₄²	—	2	6	no	—
40	C₃×C₄²	C₃⋊C₈	1	4	yes	—
41	C₃×C₄²	Dic₆	1	6	yes	—
42	C₃×C₄²	D₁₂	1	6	yes	—
43	C₂×C₃⋊C₈	—	6	12	no	—
44	C₂²×C₉	C₂×C₄⋊C₄	1	3	yes	—
45	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
46	C₂²×C₉	C₂×M₁₆	1	3	yes	—
47	C₄×C₉	—	2	6	no	—
48	C₄×C₉	C₄⋊C₈	1	12	yes	complemented
49	C₄×C₉	#127	1	1	yes	characteristic
50	C₄×C₉	C₄⋊Q₈	1	3	yes	—
51	C₂²×C₈	—	9	4	no	—
52	C₂×C₄²	Dic₉	1	3	yes	—
53	C₂×C₄²	Dic₉	1	1	yes	characteristic
54	C₂×C₄²	C₂×D₉	1	3	yes	—
55	C₄⋊C₈	—	18	12	no	—
56	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
57	C₂×C₃×C₄	—	2	6	no	—
58	C₂×C₃×C₄	C₂×C₃⋊C₈	1	6	yes	—
59	C₂×C₃×C₄	#208	1	6	yes	—
60	C₂×C₃×C₄	C₄⋊Dic₃	1	6	yes	—
61	C₂×C₃×C₄	C₄⋊Dic₃	1	6	yes	—
62	C₂×C₃×C₄	C₂×Dic₆	1	3	yes	—
63	C₂×C₃×C₄	C₂×D₁₂	1	3	yes	—
64	C₃⋊C₈	—	12	4	no	—
65	C₂×C₉	#384	1	1	yes	characteristic
66	C₂×C₉	C₂×C₄⋊C₈	1	3	yes	—
67	C₂×C₉	#423	1	3	yes	—
68	C₂²×C₄	C₂×Dic₉	1	3	yes	—
69	C₂²×C₄	C₂×Dic₉	1	3	yes	—
70	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
71	C₂×C₈	—	18	12	no	—
72	C₄²	—	2	6	no	—
73	C₄²	C₉⋊C₈	1	4	yes	complemented
74	C₄²	Dic₁₈	1	6	yes	—
75	C₄²	D₃₆	1	6	yes	—
76	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
77	C₂²×C₃	C₂×#208	1	3	yes	—
78	C₂²×C₃	C₂×C₄⋊Dic₃	1	3	yes	—
79	C₃×C₄	—	2	6	no	—
80	C₃×C₄	C₄⋊(C₃⋊C₈)	1	12	yes	—
81	C₃×C₄	#869	1	3	yes	—
82	C₃×C₄	#874	1	1	yes	characteristic
83	C₉	#1842	1	1	yes	characteristic, complemented, Sylow
84	C₂³	C₂²×Dic₉	1	1	yes	characteristic
85	C₂×C₄	—	2	6	no	—
86	C₂×C₄	C₂×C₉⋊C₈	1	6	yes	—
87	C₂×C₄	#3667	1	6	yes	—
88	C₂×C₄	C₄⋊Dic₉	1	6	yes	—
89	C₂×C₄	C₄⋊Dic₉	1	6	yes	—
90	C₂×C₄	C₂×Dic₁₈	1	3	yes	—
91	C₂×C₄	C₂×D₃₆	1	3	yes	—
92	C₈	—	36	4	no	—
93	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	3	yes	—
94	C₂×C₃	#4924	1	3	yes	—
95	C₂×C₃	#4935	1	1	yes	characteristic
96	C₂²	C₂²×C₉⋊C₈	1	1	yes	characteristic
97	C₂²	C₂×#3667	1	3	yes	—
98	C₂²	C₂×C₄⋊Dic₉	1	3	yes	—
99	C₄	—	2	6	no	—
100	C₄	C₄⋊(C₉⋊C₈)	1	12	yes	complemented
101	C₄	#63431	1	3	yes	—
102	C₄	#63436	1	1	yes	characteristic
103	C₃	#69394	1	1	yes	characteristic
104	C₂	C₂×C₄⋊(C₉⋊C₈)	1	3	yes	—
105	C₂	#10588604	1	3	yes	—
106	C₂	#10588615	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄³×C₉

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₉⋊C₈)

C₂

yes

maximal

#69394

—

maximal

C₂×C₄²×C₉

C₄

yes

—

C₂×C₄²×C₉

C₄

yes

characteristic

C₂×C₄²×C₉

C₂²

yes

—

C₂²×C₉⋊C₈

C₂²

yes

—

C₄⋊(C₉⋊C₈)

—

C₃×C₄³

D₃

yes

characteristic

C₂×C₄⋊(C₃⋊C₈)

—

C₂²×C₄×C₉

C₂×C₄

yes

—

C₂²×C₄×C₉

C₂×C₄

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₄²×C₉

—

C₄²×C₉

C₈

yes

complemented

C₄²×C₉

D₄

yes

—

C₄²×C₉

Q₈

yes

—

C₂×C₉⋊C₈

—

#1842

—

Sylow

C₂×C₃×C₄²

Dic₃

yes

—

C₂×C₃×C₄²

Dic₃

yes

characteristic

C₂×C₃×C₄²

C₂×D₃

yes

—

C₂²×C₃⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₂³×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

C₄⋊C₄

yes

—

C₂×C₄×C₉

C₄⋊C₄

yes

—

C₂×C₄×C₉

C₂×C₈

yes

—

C₂×C₄×C₉

M₁₆

yes

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×Q₈

yes

—

C₉⋊C₈

—

C₂×C₄⋊C₈

—

C₄³

D₉

yes

characteristic

C₂²×C₃×C₄

C₂×Dic₃

yes

—

C₂²×C₃×C₄

C₂×Dic₃

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₃×C₄²

—

C₃×C₄²

C₃⋊C₈

yes

—

C₃×C₄²

Dic₆

yes

—

C₃×C₄²

D₁₂

yes

—

C₂×C₃⋊C₈

—

C₂²×C₉

C₂×C₄⋊C₄

yes

—

C₂²×C₉

C₂²×C₈

yes

characteristic

C₂²×C₉

C₂×M₁₆

yes

—

C₄×C₉

—

C₄×C₉

C₄⋊C₈

yes

complemented

C₄×C₉

#127

yes

characteristic

C₄×C₉

C₄⋊Q₈

yes

—

C₂²×C₈

—

C₂×C₄²

Dic₉

yes

—

C₂×C₄²

Dic₉

yes

characteristic

C₂×C₄²

C₂×D₉

yes

—

C₄⋊C₈

—

C₂³×C₃

C₂²×Dic₃

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃⋊C₈

yes

—

C₂×C₃×C₄

#208

yes

—

C₂×C₃×C₄

C₄⋊Dic₃

yes

—

C₂×C₃×C₄

C₄⋊Dic₃

yes

—

C₂×C₃×C₄

C₂×Dic₆

yes

—

C₂×C₃×C₄

C₂×D₁₂

yes

—

C₃⋊C₈

—

C₂×C₉

#384

yes

characteristic

C₂×C₉

C₂×C₄⋊C₈

yes

—

C₂×C₉

#423

yes

—

C₂²×C₄

C₂×Dic₉

yes

—

C₂²×C₄

C₂×Dic₉

yes

—

C₂²×C₄

C₂²×D₉

yes

characteristic

C₂×C₈

—

C₄²

—

C₄²

C₉⋊C₈

yes

complemented

C₄²

Dic₁₈

yes

—

C₄²

D₃₆

yes

—

C₂²×C₃

C₂²×C₃⋊C₈

yes

characteristic

C₂²×C₃

C₂×#208

yes

—

C₂²×C₃

C₂×C₄⋊Dic₃

yes

—

C₃×C₄

—

C₃×C₄

C₄⋊(C₃⋊C₈)

yes

—

C₃×C₄

#869

yes

—

C₃×C₄

#874

yes

characteristic

C₉

#1842

yes

characteristic, complemented, Sylow

C₂³

C₂²×Dic₉

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₉⋊C₈

yes

—

C₂×C₄

#3667

yes

—

C₂×C₄

C₄⋊Dic₉

yes

—

C₂×C₄

C₄⋊Dic₉

yes

—

C₂×C₄

C₂×Dic₁₈

yes

—

C₂×C₄

C₂×D₃₆

yes

—

C₈

—

C₂×C₃

C₂×C₄⋊(C₃⋊C₈)

yes

—

C₂×C₃

#4924

yes

—

C₂×C₃

#4935

yes

characteristic

C₂²

C₂²×C₉⋊C₈

yes

characteristic

C₂²

C₂×#3667

yes

—

C₂²

C₂×C₄⋊Dic₉

yes

—

C₄

—

100

C₄

C₄⋊(C₉⋊C₈)

yes

complemented

101

C₄

#63431

yes

—

102

C₄

#63436

yes

characteristic

103

C₃

#69394

yes

characteristic

104

C₂

C₂×C₄⋊(C₉⋊C₈)

yes

—

105

C₂

#10588604

yes

—

106

C₂

#10588615

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₈
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₄³×C₉	C₂
socle, prime radical	C₂³×C₃	C₂²×Dic₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₄³	D₉
hypocenter	C₉	#1842
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center

C₂²×C₄

C₂²×D₉

commutator

C₂²×C₉

C₂²×C₈

Frattini

C₂²×C₃×C₄

C₂²×D₃

Fitting

C₄³×C₉

C₂

socle, prime radical

C₂³×C₃

C₂²×Dic₃

cosocle

C₂²×C₄×C₉

C₂³

hypercenter

C₄³

D₉

C₉

G ▷ C₂²×C₉ ▷ C₉

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
15. C₄²×C₉	C₈	semidirect
48. C₄×C₉	C₄⋊C₈	semidirect
73. C₄²	C₉⋊C₈	semidirect
83. C₉	#1842	semidirect
100. C₄	C₄⋊(C₉⋊C₈)	semidirect

H ≅ G/N

type

15. C₄²×C₉

C₈

semidirect

48. C₄×C₉

C₄⋊C₈

semidirect

73. C₄²

C₉⋊C₈

semidirect

83. C₉

#1842

semidirect

100. C₄

C₄⋊(C₉⋊C₈)

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄³×C₉	2
C₂×C₄⋊(C₉⋊C₈)	2
#69394	3

subgroup

index

C₄³×C₉

C₂×C₄⋊(C₉⋊C₈)

#69394

Subgroups — #11786177

Conjugacy classes of subgroups

Complements

Maximal subgroups